Extremal (n,m)-Graphs with Respect to Distance-Degree-Based Topological Indices
نویسندگان
چکیده
In chemical graph theory, distance-degree-based topological indices are expressions of the form ∑ u6=v F (deg(u), deg(v)), d(u, v)), where F is a function, deg(u) the degree of u, and d(u, v) the distance between u and v. Setting F to be (deg(u) + deg(v))d(u, v), deg(u)deg(v)d(u, v), (deg(u)+deg(v))d(u, v)−1, and deg(u)deg(v)d(u, v)−1, we get the degree distance index DD, the Gutman index Gut, the additively weighted Harary index HA, and the multiplicatively weighted Harary index HM , respectively. Let Gn,m be the set of connected graphs of order n and size m. It is proved that if G ∈ Gn,m, where 4 ≤ n ≤ m ≤ 2n − 4, then HA(G) ≤ (m(m + 5) + 2(n − 1)(n − 3))/2 and DD(G) ≥ (4m − n)(n − 1) − (m − n + 1)(m − n + 6). The extremal graphs are characterized in both cases and are the same. Similarly, the graphs from Gn,m with m = n+ ( k 2 ) − k, 2 ≤ k ≤ n− 1, maximizing the multiplicatively weighted Harary index and minimizing the Gutman index are obtained.
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